A constraint satisfaction programming approach for computing manufacturable stacking sequences

Highlights

• An algorithm is presented to generate stacking sequences which satisfy the manufacturing rules.

• The manufacturing rules are the blending and the design rules.

• The combinatorial nature of the problem is adressed using a constraint satisfaction approach.

• The proposed algorithm can handle complex study cases which cannot be handled with the existing methods.

• A case study shows that this algorithm can be used by the mechanical designers to define the manfacturable design space.

Abstract

An algorithm is proposed to generate stacking sequences which comply with the requirements of the composite manufacturers. These rules are the blending and the design rules. The novelty of the proposed algorithm is that it can handle a general blending scheme, where a stacking sequence can be blended with other stacking sequences and it can also be the base of others. This algorithm can have two purposes: generating a manufacturable structure given the results of a preliminary design or defining a design space of a composite structure in a design process.

Introduction

Composite structures have a growing importance in the aeronautical and automotive domains due to the weight reduction and the strengthening that they can exhibit. A composite structure can have, for example, a set of panels (like in Fig. 1) and each panel can have its own stacking sequence. The fiber orientation in each ply is one of these four conventional values: $\{-45\text{,}0\text{,}45\text{,}90\}$. The design of the stacking sequences must be such that the responses of the structure to a set of load cases do not violate some safety criterion. Moreover, the stacking sequences must be designed at the computer level such that they meet the requirements of the composite manufacturers. Otherwise, the structure cannot be manufactured.

The manufacturing rules are the design and the blending rules. The design rules define the sequence layout. They can be like, for example, having a certain number of plies per orientation, being symmetric, starting with a $\pm 45$ ply. Many papers have addressed the problem of satisfying these rules. However, these rules have been considered as the constraints of an optimization problem where the objective function is the buckling load. In [1], [2], [3], the genetic algorithms are used with the penalty method in order to satisfy these rules. In [4], [5], [6], the topology approach is used where the design rules are formulated as a penalty function of four real decision variables per ply. These two approaches, based on the penalty method, have the drawback of being unable to satisfy all the design rules at the same time due to the combinatorial nature of these constraints. In [7], the integer programming approach is used where the design rules are formulated using four binary decision variables per ply. Here, all the design rules are satisfied, but this approach is only applicable to linear objective functions.

The blending rule consists in the following. Two adjacent panels must have their stacking sequences such that one is a subset of the other. In [8], [9], [10], [11], the stacking sequence guide is used to ensure the blending between panels. The sequence of a panel is a subsequence of the stacking sequence guide. A subsequence of n plies must be the first or the last n plies of the stacking sequence guide. This assumption constitutes the limitation of the method. In [12], [13], [14], [15], a shared layer approach is used to ensure the blending. In a first step, the sequences of the panels are optimized without the blending constraint. Then, in a second step, the sequences are rearranged to find a blended structure. This second step is the drawback of the method because it does not take into account the objective function (the buckling) of the initial step. In [16], [17], a general definition for the blending is considered without any assumption on the ply drop-offs between the panels. A penalty function, based on the differences (the edit distance) between the sequences of two adjacent panels, is used in the optimization process. This approach showed that is it not efficient when it is coupled with the design rules.

In summary, the previous research has addressed the manufacturing rules using the penalty method in an optimization context. It is not an adequate approach given the combinatorial nature of all the rules. This approach makes a trade-off between satisfying the constraints and the objective function. Therefore, it does not guarantee the satisfaction of all the rules.

This paper proposes an algorithm which generates stacking sequences which comply with the blending and design rules. It can handle a blending scheme where the stacking sequence can be blended with other stacking sequences and it can also be the base of others. The advantage of this algorithm is its efficiency in satisfying all the design and blending rules, and thus generating a completely manufacturable structure. This algorithm is only dedicated to the generation of one or many stacking sequences satisfying the manufacturing rules. However it does not give compute the number of admissible stacking sequences. The algorithm does not deal with optimization. In [18] the authors have proposed a combinatorial method to optimize a buckling load based on this algorithm but for a special blending scheme (the first one in numerical experiments): one thickness is associated to one stacking sequence. This work is compared with the topology optimization of the paper [6], [19].

The general blending scheme considered in this paper is the case of many industrial applications. It also provides a catalog of stacking sequences which meets the requirements of the composite manufacturers. This catalog can be considered as the design space of a composite structure in a design process. Therefore, this paper does not focus on the mechanical response or the finite elements analysis related to a composite structure. It only concerns the combinatorial algorithm generating manufacturable stacking sequences which are the input of finite elements analysis.